Thurston (1983): classification of maximal Riemannian 3-geometries that admit a compact realization

Paolo Piccione Isometry group of Lorentz manifolds

Homogeneous Lorentz Geometries

GLie group,H ⊂Gclosed subgroup

homogeneous spaceG/H

(G,G/H)is aLorentz geometryif the action ofGonG/H preserves a Lorentzian metric tensor

A compact manifoldM islocally modeled by(G,G/H)(or, M is a realization of(G,G/H)if there exists an atlas of charts ofM taking values in open subset with transition maps inG

In this case, allG-invariantobjectsonG/Hpass toM (G,G/H)ismaximalif6 ∃G⁰ )Gacting onG/H

W. Thurston (1983): classification of maximal Riemannian 3-geometries that admit a compact realization.

Homogeneous Lorentz Geometries

GLie group,H ⊂Gclosed subgroup homogeneous spaceG/H

(G,G/H)is aLorentz geometryif the action ofGonG/H preserves a Lorentzian metric tensor

A compact manifoldM islocally modeled by(G,G/H)(or, M is a realization of(G,G/H)if there exists an atlas of charts ofM taking values in open subset with transition maps inG

In this case, allG-invariantobjectsonG/Hpass toM (G,G/H)ismaximalif6 ∃G⁰ )Gacting onG/H

W. Thurston (1983): classification of maximal Riemannian 3-geometries that admit a compact realization.

Paolo Piccione Isometry group of Lorentz manifolds

Homogeneous Lorentz Geometries

GLie group,H ⊂Gclosed subgroup homogeneous spaceG/H

(G,G/H)is aLorentz geometryif the action ofGonG/H preserves a Lorentzian metric tensor

A compact manifoldM islocally modeled by(G,G/H)(or, M is a realization of(G,G/H)if there exists an atlas of charts ofM taking values in open subset with transition maps inG

In this case, allG-invariantobjectsonG/Hpass toM (G,G/H)ismaximalif6 ∃G⁰ )Gacting onG/H

W. Thurston (1983): classification of maximal Riemannian 3-geometries that admit a compact realization.

Homogeneous Lorentz Geometries

GLie group,H ⊂Gclosed subgroup homogeneous spaceG/H

(G,G/H)is aLorentz geometryif the action ofGonG/H preserves a Lorentzian metric tensor

A compact manifoldMislocally modeled by(G,G/H)(or, M is a realization of(G,G/H)if there exists an atlas of charts ofM taking values in open subset with transition maps inG

In this case, allG-invariantobjectsonG/Hpass toM (G,G/H)ismaximalif6 ∃G⁰ )Gacting onG/H

W. Thurston (1983): classification of maximal Riemannian 3-geometries that admit a compact realization.

Paolo Piccione Isometry group of Lorentz manifolds

Homogeneous Lorentz Geometries

GLie group,H ⊂Gclosed subgroup homogeneous spaceG/H

(G,G/H)is aLorentz geometryif the action ofGonG/H preserves a Lorentzian metric tensor

A compact manifoldMislocally modeled by(G,G/H)(or, M is a realization of(G,G/H)if there exists an atlas of charts ofM taking values in open subset with transition maps inG

In this case, allG-invariantobjectsonG/Hpass toM

(G,G/H)ismaximalif6 ∃G⁰ )Gacting onG/H

W. Thurston (1983): classification of maximal Riemannian 3-geometries that admit a compact realization.

Homogeneous Lorentz Geometries

GLie group,H ⊂Gclosed subgroup homogeneous spaceG/H

(G,G/H)is aLorentz geometryif the action ofGonG/H preserves a Lorentzian metric tensor

A compact manifoldMislocally modeled by(G,G/H)(or, M is a realization of(G,G/H)if there exists an atlas of charts ofM taking values in open subset with transition maps inG

In this case, allG-invariantobjectsonG/Hpass toM (G,G/H)ismaximalif6 ∃G⁰ )Gacting onG/H

W. Thurston (1983): classification of maximal Riemannian 3-geometries that admit a compact realization.

Paolo Piccione Isometry group of Lorentz manifolds

Homogeneous Lorentz Geometries

GLie group,H ⊂Gclosed subgroup homogeneous spaceG/H

(G,G/H)is aLorentz geometryif the action ofGonG/H preserves a Lorentzian metric tensor

A compact manifoldMislocally modeled by(G,G/H)(or, M is a realization of(G,G/H)if there exists an atlas of charts ofM taking values in open subset with transition maps inG

In this case, allG-invariantobjectsonG/Hpass toM (G,G/H)ismaximalif6 ∃G⁰ )Gacting onG/H

W. Thurston (1983): classification of maximal Riemannian 3-geometries that admit a compact realization.

Examples of Lorentzian 3-geometries

Lorentz Minkowski: (R³,g_LM) = R³nO(1,2). O(1,2)

de Sitter: (dS³,g_dS) =O(1,3)/O(1,2)

By a result of E. Calabi (1963), it does not have compact realizations (only finite groups can actproperlyondSⁿ). anti de Sitter: (AdS³,g_AdS) =O(2,2)

O(1,2) Alternative description: SL(2,R)

Lorentz–Heisenberg geometry Lorentz–Sol₃geometry

Paolo Piccione Isometry group of Lorentz manifolds

Examples of Lorentzian 3-geometries

Lorentz Minkowski: (R³,g_LM) = R³nO(1,2). O(1,2)

de Sitter: (dS³,g_dS) =O(1,3)/O(1,2)

By a result of E. Calabi (1963), it does not have compact realizations (only finite groups can actproperlyondSⁿ). anti de Sitter: (AdS³,g_AdS) =O(2,2)

O(1,2) Alternative description: SL(2,R)

Lorentz–Heisenberg geometry Lorentz–Sol₃geometry

Examples of Lorentzian 3-geometries

Lorentz Minkowski: (R³,g_LM) = R³nO(1,2). O(1,2) de Sitter: (dS³,g_dS) =O(1,3)/O(1,2)

By a result of E. Calabi (1963), it does not have compact realizations (only finite groups can actproperlyondSⁿ).

anti de Sitter: (AdS³,g_AdS) =O(2,2)

O(1,2) Alternative description: SL(2,R)

Lorentz–Heisenberg geometry Lorentz–Sol₃geometry

Paolo Piccione Isometry group of Lorentz manifolds

Examples of Lorentzian 3-geometries

Lorentz Minkowski: (R³,g_LM) = R³nO(1,2). O(1,2)

de Sitter: (dS³,g_dS) =O(1,3)/O(1,2)

By a result of E. Calabi (1963), it does not have compact realizations (only finite groups can actproperlyondSⁿ).

anti de Sitter:(AdS³,g_AdS) =O(2,2) O(1,2)

Alternative description: SL(2,R) Lorentz–Heisenberg geometry Lorentz–Sol₃geometry

Examples of Lorentzian 3-geometries

Lorentz Minkowski: (R³,g_LM) = R³nO(1,2). O(1,2)

de Sitter: (dS³,g_dS) =O(1,3)/O(1,2)

By a result of E. Calabi (1963), it does not have compact realizations (only finite groups can actproperlyondSⁿ).

anti de Sitter:(AdS³,g_AdS) =O(2,2) O(1,2)

Alternative description: SL(2,R)

Lorentz–Heisenberg geometry Lorentz–Sol₃geometry

Paolo Piccione Isometry group of Lorentz manifolds

Examples of Lorentzian 3-geometries

Lorentz Minkowski: (R³,g_LM) = R³nO(1,2). O(1,2)

de Sitter: (dS³,g_dS) =O(1,3)/O(1,2)

By a result of E. Calabi (1963), it does not have compact realizations (only finite groups can actproperlyondSⁿ).

anti de Sitter:(AdS³,g_AdS) =O(2,2) O(1,2) Alternative description: SL(2,R)

Lorentz–Heisenberg geometry Lorentz–Sol₃geometry

Examples of Lorentzian 3-geometries

Lorentz Minkowski: (R³,g_LM) = R³nO(1,2). O(1,2)

de Sitter: (dS³,g_dS) =O(1,3)/O(1,2)

By a result of E. Calabi (1963), it does not have compact realizations (only finite groups can actproperlyondSⁿ).

anti de Sitter:(AdS³,g_AdS) =O(2,2) O(1,2) Alternative description: SL(2,R)

Lorentz–Heisenberg geometry

Lorentz–Sol₃geometry

Paolo Piccione Isometry group of Lorentz manifolds

Examples of Lorentzian 3-geometries

Lorentz Minkowski: (R³,g_LM) = R³nO(1,2). O(1,2)

de Sitter: (dS³,g_dS) =O(1,3)/O(1,2)

By a result of E. Calabi (1963), it does not have compact realizations (only finite groups can actproperlyondSⁿ).

anti de Sitter:(AdS³,g_AdS) =O(2,2) O(1,2) Alternative description: SL(2,R)

Lorentz–Heisenberg geometry Lorentz–Sol₃geometry

The solvable case: SL(2, R )

G=SL(2,R)semi-simple Lie group,dim(G) =3 LorentzianKilling form onsl(2,R):hA,Bi=tr(AB) bi-invariant Lorentz metricgonSL(2,R)

Iso₀ SL(2,R)

= SL(2,R)×SL(2,R).

Z ∼=O₀(2,2) Z group generated by(−I₂,−I₂)

SL(2,R),g

isometric toAdS³

Γ⊂SL(2,R)co-compact lattice (i.e., discrete subgroup withM=G/Γcompact), then(M,g)is a compact homogeneous Lorentz manifold

Iso₀(M,g) =PSL(2,R)(non compact)

Paolo Piccione Isometry group of Lorentz manifolds

The solvable case: SL(2, R )

G=SL(2,R)semi-simple Lie group,dim(G) =3

LorentzianKilling form onsl(2,R):hA,Bi=tr(AB) bi-invariant Lorentz metricgonSL(2,R)

Iso₀ SL(2,R)

= SL(2,R)×SL(2,R).

Z ∼=O₀(2,2) Z group generated by(−I₂,−I₂)

SL(2,R),g

isometric toAdS³

Γ⊂SL(2,R)co-compact lattice (i.e., discrete subgroup withM=G/Γcompact), then(M,g)is a compact homogeneous Lorentz manifold

Iso₀(M,g) =PSL(2,R)(non compact)

The solvable case: SL(2, R )

G=SL(2,R)semi-simple Lie group,dim(G) =3 LorentzianKilling form onsl(2,R):hA,Bi=tr(AB)

bi-invariant Lorentz metricgonSL(2,R) Iso₀ SL(2,R)

= SL(2,R)×SL(2,R).

Z ∼=O₀(2,2) Z group generated by(−I₂,−I₂)

SL(2,R),g

isometric toAdS³

Γ⊂SL(2,R)co-compact lattice (i.e., discrete subgroup withM=G/Γcompact), then(M,g)is a compact homogeneous Lorentz manifold

Iso₀(M,g) =PSL(2,R)(non compact)

Paolo Piccione Isometry group of Lorentz manifolds

The solvable case: SL(2, R )

G=SL(2,R)semi-simple Lie group,dim(G) =3 LorentzianKilling form onsl(2,R):hA,Bi=tr(AB) bi-invariant Lorentz metricgonSL(2,R)

Iso₀ SL(2,R)

= SL(2,R)×SL(2,R).

Z ∼=O₀(2,2) Z group generated by(−I₂,−I₂)

SL(2,R),g

isometric toAdS³

Γ⊂SL(2,R)co-compact lattice (i.e., discrete subgroup withM=G/Γcompact), then(M,g)is a compact homogeneous Lorentz manifold

Iso₀(M,g) =PSL(2,R)(non compact)

The solvable case: SL(2, R )

G=SL(2,R)semi-simple Lie group,dim(G) =3 LorentzianKilling form onsl(2,R):hA,Bi=tr(AB) bi-invariant Lorentz metricgonSL(2,R)

Iso0 SL(2,R)

= SL(2,R)×SL(2,R).

Z ∼=O0(2,2) Z group generated by(−I₂,−I₂)

SL(2,R),g

isometric toAdS³

Γ⊂SL(2,R)co-compact lattice (i.e., discrete subgroup withM=G/Γcompact), then(M,g)is a compact homogeneous Lorentz manifold

Iso₀(M,g) =PSL(2,R)(non compact)

Paolo Piccione Isometry group of Lorentz manifolds

The solvable case: SL(2, R )

G=SL(2,R)semi-simple Lie group,dim(G) =3 LorentzianKilling form onsl(2,R):hA,Bi=tr(AB) bi-invariant Lorentz metricgonSL(2,R)

Iso0 SL(2,R)

= SL(2,R)×SL(2,R).

Z ∼=O0(2,2) Z group generated by(−I₂,−I₂)

SL(2,R),g

isometric toAdS³

Γ⊂SL(2,R)co-compact lattice (i.e., discrete subgroup withM=G/Γcompact), then(M,g)is a compact homogeneous Lorentz manifold

Iso₀(M,g) =PSL(2,R)(non compact)

The solvable case: SL(2, R )

G=SL(2,R)semi-simple Lie group,dim(G) =3 LorentzianKilling form onsl(2,R):hA,Bi=tr(AB) bi-invariant Lorentz metricgonSL(2,R)

Iso0 SL(2,R)

= SL(2,R)×SL(2,R).

Z ∼=O0(2,2) Z group generated by(−I₂,−I₂)

SL(2,R),g

isometric toAdS³

Γ⊂SL(2,R)co-compact lattice (i.e., discrete subgroup withM=G/Γcompact), then(M,g)is a compact homogeneous Lorentz manifold

Iso₀(M,g) =PSL(2,R)(non compact)

Paolo Piccione Isometry group of Lorentz manifolds

The solvable case: SL(2, R )

G=SL(2,R)semi-simple Lie group,dim(G) =3 LorentzianKilling form onsl(2,R):hA,Bi=tr(AB) bi-invariant Lorentz metricgonSL(2,R)

Iso0 SL(2,R)

= SL(2,R)×SL(2,R).

Z ∼=O0(2,2) Z group generated by(−I₂,−I₂)

SL(2,R),g

isometric toAdS³

Γ⊂SL(2,R)co-compact lattice (i.e., discrete subgroup withM=G/Γcompact), then(M,g)is a compact homogeneous Lorentz manifold

Iso₀(M,g) =PSL(2,R)(non compact)

The nilpotent case: Heis

³

G=Heis³= n





1 x y 0 1 z 0 0 1



:x,y,z ∈R o

heis³=span{X,Y,Z},X ∈z,[Z,Y] =X

up to automorphisms,∃three left-invariant Lorentz metrics onHeis³:

g(X,X) =0−→flatmetric

g(X,X) =−1,X ⊥span{Y,Z}=⇒Riemannian-like

g(X,X) =1,X orthogonal to the Lorentz plane spanned by Y andZ,g(Y,Z) =1=⇒Lorentz–Heisenberg geometry

Isometry group: RnHeis³(4-dim)

R: 1-parameter group of automorphism that fixX. maximal geometry

Paolo Piccione Isometry group of Lorentz manifolds

The nilpotent case: Heis

³

G=Heis³= n





1 x y 0 1 z 0 0 1



:x,y,z ∈R o

heis³=span{X,Y,Z},X ∈z,[Z,Y] =X

up to automorphisms,∃three left-invariant Lorentz metrics onHeis³:

g(X,X) =0−→flatmetric

g(X,X) =−1,X ⊥span{Y,Z}=⇒Riemannian-like

g(X,X) =1,X orthogonal to the Lorentz plane spanned by Y andZ,g(Y,Z) =1=⇒Lorentz–Heisenberg geometry

Isometry group: RnHeis³(4-dim)

R: 1-parameter group of automorphism that fixX. maximal geometry

The nilpotent case: Heis

³

G=Heis³= n





1 x y 0 1 z 0 0 1



:x,y,z ∈R o

heis³=span{X,Y,Z},X ∈z,[Z,Y] =X

up to automorphisms,∃three left-invariant Lorentz metrics onHeis³:

g(X,X) =0−→flatmetric

g(X,X) =−1,X ⊥span{Y,Z}=⇒Riemannian-like

g(X,X) =1,X orthogonal to the Lorentz plane spanned by Y andZ,g(Y,Z) =1=⇒Lorentz–Heisenberg geometry

Isometry group: RnHeis³(4-dim)

R: 1-parameter group of automorphism that fixX. maximal geometry

Paolo Piccione Isometry group of Lorentz manifolds

The nilpotent case: Heis

³

G=Heis³= n





1 x y 0 1 z 0 0 1



:x,y,z ∈R o

heis³=span{X,Y,Z},X ∈z,[Z,Y] =X

up to automorphisms,∃three left-invariant Lorentz metrics onHeis³:

g(X,X) =0−→flatmetric

g(X,X) =−1,X ⊥span{Y,Z}=⇒Riemannian-like

g(X,X) =1,X orthogonal to the Lorentz plane spanned by Y andZ,g(Y,Z) =1=⇒Lorentz–Heisenberg geometry Isometry group: RnHeis³(4-dim)

R: 1-parameter group of automorphism that fixX. maximal geometry

The nilpotent case: Heis

³

G=Heis³= n





1 x y 0 1 z 0 0 1



:x,y,z ∈R o

heis³=span{X,Y,Z},X ∈z,[Z,Y] =X

up to automorphisms,∃three left-invariant Lorentz metrics onHeis³:

g(X,X) =0−→flatmetric

g(X,X) =−1,X ⊥span{Y,Z}=⇒Riemannian-like

g(X,X) =1,X orthogonal to the Lorentz plane spanned by Y andZ,g(Y,Z) =1=⇒Lorentz–Heisenberg geometry Isometry group: RnHeis³(4-dim)

R: 1-parameter group of automorphism that fixX. maximal geometry

Paolo Piccione Isometry group of Lorentz manifolds

The nilpotent case: Heis

³

G=Heis³= n





1 x y 0 1 z 0 0 1



:x,y,z ∈R o

heis³=span{X,Y,Z},X ∈z,[Z,Y] =X

up to automorphisms,∃three left-invariant Lorentz metrics onHeis³:

g(X,X) =0−→flatmetric

g(X,X) =−1,X ⊥span{Y,Z}=⇒Riemannian-like

g(X,X) =1,X orthogonal to the Lorentz plane spanned by Y andZ,g(Y,Z) =1=⇒Lorentz–Heisenberg geometry Isometry group: RnHeis³(4-dim)

R: 1-parameter group of automorphism that fixX. maximal geometry

The nilpotent case: Heis

³

G=Heis³= n





1 x y 0 1 z 0 0 1



:x,y,z ∈R o

heis³=span{X,Y,Z},X ∈z,[Z,Y] =X

up to automorphisms,∃three left-invariant Lorentz metrics onHeis³:

g(X,X) =0−→flatmetric

g(X,X) =−1,X ⊥span{Y,Z}=⇒Riemannian-like g(X,X) =1,Xorthogonal to the Lorentz plane spanned by Y andZ,g(Y,Z) =1=⇒Lorentz–Heisenberg geometry

Isometry group: RnHeis³(4-dim)

R: 1-parameter group of automorphism that fixX. maximal geometry

Paolo Piccione Isometry group of Lorentz manifolds

The nilpotent case: Heis

³

G=Heis³= n





1 x y 0 1 z 0 0 1



:x,y,z ∈R o

heis³=span{X,Y,Z},X ∈z,[Z,Y] =X

up to automorphisms,∃three left-invariant Lorentz metrics onHeis³:

g(X,X) =0−→flatmetric

g(X,X) =−1,X ⊥span{Y,Z}=⇒Riemannian-like g(X,X) =1,Xorthogonal to the Lorentz plane spanned by Y andZ,g(Y,Z) =1=⇒Lorentz–Heisenberg geometry Isometry group: RnHeis³(4-dim)

R: 1-parameter group of automorphism that fixX.

maximal geometry

The nilpotent case: Heis

³

G=Heis³= n





1 x y 0 1 z 0 0 1



:x,y,z ∈R o

heis³=span{X,Y,Z},X ∈z,[Z,Y] =X

up to automorphisms,∃three left-invariant Lorentz metrics onHeis³:

g(X,X) =0−→flatmetric

g(X,X) =−1,X ⊥span{Y,Z}=⇒Riemannian-like g(X,X) =1,Xorthogonal to the Lorentz plane spanned by Y andZ,g(Y,Z) =1=⇒Lorentz–Heisenberg geometry Isometry group: RnHeis³(4-dim)

R: 1-parameter group of automorphism that fixX. maximal geometry

Paolo Piccione Isometry group of Lorentz manifolds

The solvable case: Sol

³

G=Sol³= n

e^−z 0 x 0 e^z y 0 0 1

x,y,z ∈R o 3-dim solvable group

sol³=span{X,Y,Z},[X,Y] =0,[X,Z] =X,[Y,Z] =−Y [sol³,sol³] =span{X,Y}

g left-invariant Lorentz metric onG:span{X,Y}=X^⊥ up to automorphisms, can assumeg(Y,Y) =1, g(Y,Z) =0,g(X,Z) =1

direct computation: R(Y,Z) =





0 −2 0

0 0 2

0 0 0



(non flat!) isometry group: RnHeis³,dim=4!!!

Obs.: Gadmits also left invariantflatmetrics maximal geometry

The solvable case: Sol

³

G=Sol³= n

e^−z 0 x 0 e^z y 0 0 1

x,y,z ∈R o 3-dim solvable group

sol³=span{X,Y,Z},[X,Y] =0,[X,Z] =X,[Y,Z] =−Y [sol³,sol³] =span{X,Y}

g left-invariant Lorentz metric onG:span{X,Y}=X^⊥ up to automorphisms, can assumeg(Y,Y) =1, g(Y,Z) =0,g(X,Z) =1

direct computation: R(Y,Z) =





0 −2 0

0 0 2

0 0 0



(non flat!) isometry group: RnHeis³,dim=4!!!

Obs.: Gadmits also left invariantflatmetrics maximal geometry

Paolo Piccione Isometry group of Lorentz manifolds

The solvable case: Sol

³

G=Sol³= n

e^−z 0 x 0 e^z y 0 0 1

x,y,z ∈R o 3-dim solvable group

sol³=span{X,Y,Z},[X,Y] =0,[X,Z] =X,[Y,Z] =−Y

[sol³,sol³] =span{X,Y}

g left-invariant Lorentz metric onG:span{X,Y}=X^⊥ up to automorphisms, can assumeg(Y,Y) =1, g(Y,Z) =0,g(X,Z) =1

direct computation: R(Y,Z) =





0 −2 0

0 0 2

0 0 0



(non flat!) isometry group: RnHeis³,dim=4!!!

Obs.: Gadmits also left invariantflatmetrics maximal geometry

The solvable case: Sol

³

G=Sol³= n

e^−z 0 x 0 e^z y 0 0 1

x,y,z ∈R o 3-dim solvable group

sol³=span{X,Y,Z},[X,Y] =0,[X,Z] =X,[Y,Z] =−Y [sol³,sol³] =span{X,Y}

g left-invariant Lorentz metric onG:span{X,Y}=X^⊥ up to automorphisms, can assumeg(Y,Y) =1, g(Y,Z) =0,g(X,Z) =1

direct computation: R(Y,Z) =





0 −2 0

0 0 2

0 0 0



(non flat!) isometry group: RnHeis³,dim=4!!!

Obs.: Gadmits also left invariantflatmetrics maximal geometry

Paolo Piccione Isometry group of Lorentz manifolds

The solvable case: Sol

³

G=Sol³= n

e^−z 0 x 0 e^z y 0 0 1

x,y,z ∈R o 3-dim solvable group

sol³=span{X,Y,Z},[X,Y] =0,[X,Z] =X,[Y,Z] =−Y [sol³,sol³] =span{X,Y}

g left-invariant Lorentz metric onG:span{X,Y}=X^⊥

up to automorphisms, can assumeg(Y,Y) =1, g(Y,Z) =0,g(X,Z) =1

direct computation: R(Y,Z) =





0 −2 0

0 0 2

0 0 0



(non flat!) isometry group: RnHeis³,dim=4!!!

Obs.: Gadmits also left invariantflatmetrics maximal geometry

The solvable case: Sol

³

G=Sol³= n

e^−z 0 x 0 e^z y 0 0 1

x,y,z ∈R o 3-dim solvable group

sol³=span{X,Y,Z},[X,Y] =0,[X,Z] =X,[Y,Z] =−Y [sol³,sol³] =span{X,Y}

g left-invariant Lorentz metric onG:span{X,Y}=X^⊥ up to automorphisms, can assumeg(Y,Y) =1, g(Y,Z) =0,g(X,Z) =1

direct computation: R(Y,Z) =





0 −2 0

0 0 2

0 0 0



(non flat!) isometry group: RnHeis³,dim=4!!!

Obs.: Gadmits also left invariantflatmetrics maximal geometry

Paolo Piccione Isometry group of Lorentz manifolds

The solvable case: Sol

³

G=Sol³= n

e^−z 0 x 0 e^z y 0 0 1

x,y,z ∈R o 3-dim solvable group

sol³=span{X,Y,Z},[X,Y] =0,[X,Z] =X,[Y,Z] =−Y [sol³,sol³] =span{X,Y}

g left-invariant Lorentz metric onG:span{X,Y}=X^⊥ up to automorphisms, can assumeg(Y,Y) =1, g(Y,Z) =0,g(X,Z) =1

direct computation: R(Y,Z) =





0 −2 0

0 0 2

0 0 0



(non flat!)

isometry group: RnHeis³,dim=4!!!

Obs.: Gadmits also left invariantflatmetrics maximal geometry

The solvable case: Sol

³

G=Sol³= n

e^−z 0 x 0 e^z y 0 0 1

x,y,z ∈R o 3-dim solvable group

sol³=span{X,Y,Z},[X,Y] =0,[X,Z] =X,[Y,Z] =−Y [sol³,sol³] =span{X,Y}

g left-invariant Lorentz metric onG:span{X,Y}=X^⊥ up to automorphisms, can assumeg(Y,Y) =1, g(Y,Z) =0,g(X,Z) =1

direct computation: R(Y,Z) =





0 −2 0

0 0 2

0 0 0



(non flat!) isometry group: RnHeis³,dim=4!!!

Obs.: Gadmits also left invariantflatmetrics maximal geometry

Paolo Piccione Isometry group of Lorentz manifolds

The solvable case: Sol

³

G=Sol³= n

e^−z 0 x 0 e^z y 0 0 1

x,y,z ∈R o 3-dim solvable group

sol³=span{X,Y,Z},[X,Y] =0,[X,Z] =X,[Y,Z] =−Y [sol³,sol³] =span{X,Y}

g left-invariant Lorentz metric onG:span{X,Y}=X^⊥ up to automorphisms, can assumeg(Y,Y) =1, g(Y,Z) =0,g(X,Z) =1

direct computation: R(Y,Z) =





0 −2 0

0 0 2

0 0 0



(non flat!) isometry group: RnHeis³,dim=4!!!

Obs.: Gadmits also left invariantflatmetrics

maximal geometry

The solvable case: Sol

³

G=Sol³= n

e^−z 0 x 0 e^z y 0 0 1

x,y,z ∈R o 3-dim solvable group

sol³=span{X,Y,Z},[X,Y] =0,[X,Z] =X,[Y,Z] =−Y [sol³,sol³] =span{X,Y}

g left-invariant Lorentz metric onG:span{X,Y}=X^⊥ up to automorphisms, can assumeg(Y,Y) =1, g(Y,Z) =0,g(X,Z) =1

direct computation: R(Y,Z) =





0 −2 0

0 0 2

0 0 0



(non flat!) isometry group: RnHeis³,dim=4!!!

Obs.: Gadmits also left invariantflatmetrics maximal geometry

Paolo Piccione Isometry group of Lorentz manifolds

No documento Short course On the isometry group of Lorentz manifolds (páginas 157-200)